Can digital computers understand infinity?
I think this is a fairly common misconception about AI and computers, especially among laypeople. There are several things to unpack here.
Let's suppose that there's something special about infinity (or about continuous concepts) that makes them especially difficult for AI. For this to be true, it must both be the case that humans can understand these concepts while they remain alien to machines, and that there exist other concepts that are not like infinity that both humans and machines can understand. What I'm going to show in this answer is that wanting both of these things leads to a contradiction.
The root of this misunderstanding is the problem of what it means to understand. Understanding is a vague term in everyday life, and that vague nature contributes to this misconception.
If by understanding, we mean that a computer has the conscious experience of a concept, then we quickly become trapped in metaphysics. There is a long running, and essentially open debate about whether computers can &quot;understand&quot; anything in this sense, and even at times, about whether humans can! You might as well ask whether a computer can &quot;understand&quot; that 2+2=4. Therefore, if there's something special about understanding infinity, it cannot be related to &quot;understanding&quot; in the sense of subjective experience.
So, let's suppose that by &quot;understand&quot;, we have some more specific definition in mind. Something that would make a concept like infinity more complicated for a computer to &quot;understand&quot; than a concept like arithmetic. Our more concrete definition for &quot;understanding&quot; must relate to some objectively measurable capacity or ability related to the concept (otherwise, we're back in the land of subjective experience). Let's consider what capacity or ability might we pick that would make infinity a special concept, understood by humans and not machines, unlike say, arithmetic.
We might say that a computer (or a person) understands a concept if it can provide a correct definition of that concept. However, if even one human understands infinity by this definition, then it should be easy for them to write down the definition. Once the definition is written down, a computer program can output it. Now the computer &quot;understands&quot; infinity too. This definition doesn't work for our purposes.
We might say that an entity understands a concept if it can apply the concept correctly. Again, if even the one person understands how to apply the concept of infinity correctly, then we only need to record the rules they are using to reason about the concept, and we can write a program that reproduces the behavior of this system of rules. Infinity is actually very well characterized as a concept, captured in ideas like Aleph Numbers. It is not impractical to encode these systems of rules in a computer, at least up to the level that any human understands them. Therefore, computers can &quot;understand&quot; infinity up to the same level of understanding as humans by this definition as well. So this definition doesn't work for our purposes.
We might say that an entity &quot;understands&quot; a concept if it can logically relate that concept to arbitrary new ideas. This is probably the strongest definition, but we would need to be pretty careful here: very few humans (proportionately) have a deep understanding of a concept like infinity. Even fewer can readily relate it to arbitrary new concepts. Further, algorithms like the General Problem Solver can, in principal, derive any logical consequences from a given body of facts, given enough time. Perhaps under this definition computers understand infinity better than most humans, and there is certainly no reason to suppose that our existing algorithms will not further improve this capability over time. This definition does not seem to meet our requirements either.
Finally, we might say that an entity &quot;understands&quot; a concept if it can generate examples of it. For example, I can generate examples of problems in arithmetic, and their solutions. Under this definition, I probably do not &quot;understand&quot; infinity, because I cannot actually point to or create any concrete thing in the real world that is definitely infinite. I cannot, for instance, actually write down an infinitely long list of numbers, merely formulas that express ways to create ever longer lists by investing ever more effort in writing them out. A computer ought to be at least as good as me at this. This definition also does not work.
This is not an exhaustive list of possible definitions of &quot;understands&quot;, but we have covered &quot;understands&quot; as I understand it pretty well. Under every definition of understanding, there isn't anything special about infinity that separates it from other mathematical concepts.
So the upshot is that, either you decide a computer doesn't &quot;understand&quot; anything at all, or there's no particularly good reason to suppose that infinity is harder to understand than other logical concepts. If you disagree, you need to provide a concrete definition of &quot;understanding&quot; that does separate understanding of infinity from other concepts, and that doesn't depend on subjective experiences (unless you want to claim your particular metaphysical views are universally correct, but that's a hard argument to make).
Infinity has a sort of semi-mystical status among the lay public, but it's really just like any other mathematical system of rules: if we can write down the rules by which infinity operates, a computer can do them as well as a human can (or better).